> Are they the _only_ flavour of curve that has a nice geometric group law?

For affine conics over the real numbers, the non-degenerate ones are ellipses (affine transform to complex unit circle), hyperbolas (affine transform to y=1/x and use the group law (x,y)(x',y')=(xx',yy')) and parabolas (affine transform to y=x^2 and use (x,y)(x',y')=(xx',yy')).

I was thinking about projective conics, but it turns out there are no algebraic group laws on those, because they're always ill-defined over an algebraically-closed field. Moreover, over the reals and other non-algebraically closed fields k, the definition of a "regular map" needs to consider points with coordinates taking values in the algebraic closure of k.